Random access test as an identifier of nonclassicality*

Random access codes (RACs) are an intriguing class of communication tasks that reveal an operational and quantitative difference between classical and quantum information processing. We formulate a natural generalization of RACs and call them random access tests (RATs), defined for any finite collection of measurements in an arbitrary finite dimensional general probabilistic theory. These tests can be used to examine collective properties of collections of measurements. We show that the violation of a classical bound in a RAT is a signature of either measurement incompatibility or super information storability. The polygon theories are exhaustively analysed and a critical difference between even and odd polygon theories is revealed.